Fast Fourier Solutions

Subject:	Fast Fourier Solutions
Date:	Sat, 5 Dec 2009 18:03:39 -0800 (PST)

== Solutions ==
[[FT1|solution]] Find the [[Fourier transform]] of

f(t) = e^{-|t|}\,

[[FT2|solution]] Find the Fourier transform of

f(t) = \begin{cases}1&|t|<1\\0&|t|>1\end{cases}\,

[[PDEFT1|
solution]]

u_t=ku_{xx}\,

width=50%>

u(0,t) = 0\,

u(x,0) = f(x)\,

t>0,\,\,0

[[PDEFT2|
solution]]

u_{xx}+u_{yy}=0\,

width=50%>

u(0,y) = 0\,

u(1,y) = 0\,

u(x,0) = 0\,

u(x,1) = B x(1-x)\,

t>0,\,\,0 td>

[[PDEFT3|
solution]]

u_t=-u_{xxxx}\,

width=50%>

u(x,0) = f(x)\,

t>0,\,\,x\isin\mathbb{R},

table>

[[PDEFT4|
solution]]

u_{tt}=c^2\,u_{xx}\,

width=50%>

u(x,0) = f(x)\,

u_t(x,0)=g(x)\,

t>0,\,\,x\isin\mathbb{R},

[[PDEFT5|
solution]]

u_{xx}+u_{yy}+u_{zz}=0\,

width=50%>

u(x,y,0) = f(x,y)\,

Auxiliary condition:

u

is bounded.

t>0,\,\,x,y\isin\mathbb{R},\,\,\,z>0\,

u(x,y,z) = \int\!\!\!\int_\Re e^{i \lambda x + i \mu y} B(\lambda,\mu) e^{-\sqrt{\lambda^2+\mu^2}\,z}\,d\lambda d\mu\,

[Quick Answer] Write the form of the solution:

u_{tt}=c^2(u_{xx}+u_{yy})\,

u(x,0,t) = g(x)\,

u(0,y,t) = h(y)\,

u(x,y,0) = 0\,

u_t(x,y,0) = f(x,y)\,

0 0\,

u(x,y,t) = \int_0^\infty \int_0^\infty U(\lambda,\mu,t) \sin(\lambda x) \sin(\mu y)\,d\lambda\,d\mu\,

[Quick Answer] Write the form of the solution:

u_{tt}=c^2(u_{xx}+u_{yy})\,

u_y(x,0,t) = g(x)\,

u(0,y,t) = h(y)\,

u(x,y,0) = 0\,

u_t(x,y,0) = f(x,y)\,

0 0\,

u(x,y,t) = \int_0^\infty \int_0^\infty U(\lambda,\mu,t) \sin(\lambda x) \cos(\mu y)\,d\lambda\,d\mu\,

valign=top width=50%>

u_y(x,0,t) = g(x)\,

u(0,y,t) = h(y)\,

u(x,y,0) = 0\,

u_t(x,y,0) = f(x,y)\,

0 0\,

[[PDEFT6|solution]]

u_{tt}=c^2(u_{xx}+u_{yy})\,

u(x,y,t) = \int_0^\infty \int_0^\infty U(\lambda,\mu,t) \sin(\lambda x) \cos(\mu y)\,d\lambda\,d\mu\,

[[Partial Differential Equations]]

[[Main Page]]

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M. Michael Musatov